Question: Simplify and expand the following expression: $ \dfrac{1}{5n - 50}- \dfrac{3}{2n - 4}- \dfrac{4n}{n^2 - 12n + 20} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{1}{5n - 50} = \dfrac{1}{5(n - 10)}$ We can factor a $2$ out of denominator in the second term: $ \dfrac{3}{2n - 4} = \dfrac{3}{2(n - 2)}$ We can factor the quadratic in the third term: $ \dfrac{4n}{n^2 - 12n + 20} = \dfrac{4n}{(n - 10)(n - 2)}$ Now we have: $ \dfrac{1}{5(n - 10)}- \dfrac{3}{2(n - 2)}- \dfrac{4n}{(n - 10)(n - 2)} $ The least common multiple of the denominators is: $ 10(n - 10)(n - 2)$ In order to get the first term over $10(n - 10)(n - 2)$ , multiply by $\dfrac{2(n - 2)}{2(n - 2)}$ $ \dfrac{1}{5(n - 10)} \times \dfrac{2(n - 2)}{2(n - 2)} = \dfrac{2(n - 2)}{10(n - 10)(n - 2)} $ In order to get the second term over $10(n - 10)(n - 2)$ , multiply by $\dfrac{5(n - 10)}{5(n - 10)}$ $ \dfrac{3}{2(n - 2)} \times \dfrac{5(n - 10)}{5(n - 10)} = \dfrac{15(n - 10)}{10(n - 10)(n - 2)} $ In order to get the third term over $10(n - 10)(n - 2)$ , multiply by $\dfrac{10}{10}$ $ \dfrac{4n}{(n - 10)(n - 2)} \times \dfrac{10}{10} = \dfrac{40n}{10(n - 10)(n - 2)} $ Now we have: $ \dfrac{2(n - 2)}{10(n - 10)(n - 2)} - \dfrac{15(n - 10)}{10(n - 10)(n - 2)} - \dfrac{40n}{10(n - 10)(n - 2)} $ $ = \dfrac{ 2(n - 2) - 15(n - 10) - 40n} {10(n - 10)(n - 2)} $ Expand: $ = \dfrac{2n - 4 - 15n + 150 - 40n}{10n^2 - 120n + 200} $ $ = \dfrac{-53n + 146}{10n^2 - 120n + 200}$